Teukolsky on slowly-rotating Kerr-de Sitter in the vanishing $\Lambda$ limit

We prove energy, Morawetz and $r^p$-weighted estimates for solutions to the Teukolsky equation set on a slowly-rotating Kerr-de Sitter background. The main feature of our estimates is their uniformity with respect to the cosmological constant $\Lambda>0$ (thus allowed to tend to $0$), while they hold on the whole domain of outer communications, extending up to $r\sim \Lambda^{-\frac{1}{2}}$. As an application of our result, we recover well-known corresponding estimates for solutions to Teukolsky on a slowly-rotating Kerr background in the limit $\Lambda\to 0$.